If $y = \tan^{-1}\left(\sqrt{\dfrac{1-\cos x}{1+\cos x}}\right)$, then $\dfrac{dy}{dx}$ equals:
A$\cos x$
B$1$
C$1/2$
D$\sin x / 2$
Answer & Solution
Correct answer: C. $1/2$
Use $\sqrt{(1-\cos x)/(1+\cos x)} = \tan(x/2)$ (half-angle identity). So $y = \tan^{-1}\tan(x/2) = x/2$. Hence $dy/dx = 1/2$. Beautiful simplification!
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